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314
Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices
, 2008
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Two linear transformations each tridiagonal with respect to an eigenbasis of the other; comments on the split decomposition
, 2003
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Advanced determinant calculus: a complement
 Linear Algebra Appl
"... Abstract. This is a complement to my previous article “Advanced Determinant Calculus ” (Séminaire Lotharingien Combin. 42 (1999), Article B42q, 67 pp.). In the present article, I share with the reader my experience of applying the methods described in the previous article in order to solve a particu ..."
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Cited by 49 (6 self)
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Abstract. This is a complement to my previous article “Advanced Determinant Calculus ” (Séminaire Lotharingien Combin. 42 (1999), Article B42q, 67 pp.). In the present article, I share with the reader my experience of applying the methods described in the previous article in order to solve a particular problem from number theory (G. Almkvist, J. Petersson and the author, Experiment. Math. 12 (2003), 441– 456). Moreover, I add a list of determinant evaluations which I consider as interesting, which have been found since the appearance of the previous article, or which I failed to mention there, including several conjectures and open problems. 1.
Orthogonal polynomials for exponential weights x2ρe−2Q(x) on [0,d
 J. Approx. Theory
"... xi + 476 pp With each of a large class of positive measures µ on the real line it is possible to associate a sequence {pn(x)} of orthogonal polynomials with the property that pm(x)pn(x)dµ(x) = 0, m ̸ = n, 1, m = n. (1) Such a sequence satisfies a three term recurrence relation xpn(x) = anPn+1(x) + ..."
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Cited by 47 (15 self)
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xi + 476 pp With each of a large class of positive measures µ on the real line it is possible to associate a sequence {pn(x)} of orthogonal polynomials with the property that pm(x)pn(x)dµ(x) = 0, m ̸ = n, 1, m = n. (1) Such a sequence satisfies a three term recurrence relation xpn(x) = anPn+1(x) + bnPn(x) + an−1Pn−1(x) (2) Conversely, for suitable starting values and coefficient sequences {an}, {bn}, the recurrence relation (2) generates a sequence of polynomials satisfying (1) for some measure µ. The polynomials have their zeros within the interval of support of the measure. Examples date from the 19th century. The Jacobi polynomials P (α,β) n (x) are orthogonal with respect to µ with support [−1, 1], where dµ = (1 − x) α+1 (1 +
Distribution on partitions, point processes, and the hypergeometric kernel
 Comm. Math. Phys
"... Abstract. We study a 3–parametric family of stochastic point processes on the one–dimensional lattice originated from a remarkable family of representations of the infinite symmetric group. We prove that the correlation functions of the processes are given by determinantal formulas with a certain ke ..."
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Cited by 47 (17 self)
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Abstract. We study a 3–parametric family of stochastic point processes on the one–dimensional lattice originated from a remarkable family of representations of the infinite symmetric group. We prove that the correlation functions of the processes are given by determinantal formulas with a certain kernel. The kernel can be expressed through the Gauss hypergeometric function; we call it the hypergeometric kernel. In a scaling limit our processes approximate the processes describing the decomposition of representations mentioned above into irreducibles. As we showed before, see math.RT/9810015, the correlation functions of these limit processes also have determinantal form with so–called Whittaker kernel. We show that the scaling limit of the hypergeometric kernel is the Whittaker kernel. The integral operator corresponding to the Whittaker kernel is an integrable operator as defined by Its, Izergin, Korepin, and Slavnov. We argue that the hypergeometric kernel can be considered as a kernel defining a ‘discrete integrable operator’. We also show that the hypergeometric kernel degenerates for certain values of parameters to the Christoffel–Darboux kernel for Meixner orthogonal polynomials.
The arctic circle boundary and the Airy process
 Ann. Prob
, 2005
"... Abstract. We prove that the, appropriately rescaled, boundary of the north polar region in the Aztec diamond converges to the Airy process. The proof uses certain determinantal point processes given by the extended Krawtchouk kernel. We also prove a version of Propp’s conjecture concerning the struc ..."
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Cited by 42 (1 self)
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Abstract. We prove that the, appropriately rescaled, boundary of the north polar region in the Aztec diamond converges to the Airy process. The proof uses certain determinantal point processes given by the extended Krawtchouk kernel. We also prove a version of Propp’s conjecture concerning the structure of the tiling at the center of the Aztec diamond. 1. Introduction and
The Asymptotic Zero Distribution of Orthogonal Polynomials With Varying Recurrence Coefficients
"... this paper to ll this gap. To state our theorem we use the notation lim n=N!t X n;N = X 4 Kuijlaars and Van Assche to denote the property that in the doubly indexed sequence X n;N we have lim j!1 X n j ;N j = X whenever n j and N j are two sequences of natural numbers such that N j ! 1 and n j ..."
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Cited by 40 (9 self)
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this paper to ll this gap. To state our theorem we use the notation lim n=N!t X n;N = X 4 Kuijlaars and Van Assche to denote the property that in the doubly indexed sequence X n;N we have lim j!1 X n j ;N j = X whenever n j and N j are two sequences of natural numbers such that N j ! 1 and n j =N j ! t as j !1. For example, the convergence in Proposition 1.3 may be expressed by lim n=N!t (p n;N ) = w;t : We will use this notation throughout the rest of the paper. Our main result is the following. Theorem 1.4 Let for each N 2 N, two sequences fa n;N g 1 n=1 , a n;N > 0 and fb n;N g 1 n=0 of recurrence coecients be given, together with orthogonal polynomials p n;N generated by the recurrence xp n;N (x) = a n+1;N p n+1;N (x) + b n;N p n;N (x) + a n;N p n 1;N (x); n 0; (1.6) and the initial conditions p 0;N 1 and p 1;N 0. Suppose that there exist two continuous functions a : (0; 1) ! [0; 1), b : (0; 1) ! R, such that lim n=N!t a n;N = a(t); lim n=N!t b n;N = b(t) (1.7) whenever t > 0. Dene the functions (t) := b(t) 2a(t); (t) := b(t) + 2a(t); t > 0: (1.8) Then we have for every t > 0, lim n=N!t (p n;N ) = 1 t Z t 0 ! [(s);(s)] ds: (1.9) Here ! [;] is the measure given by (1.4) if < . If = , then ! [;] is the Dirac point mass at . Remark 1.5 The measure on the righthand side of (1.9) is the average of the equilibrium measures of the varying intervals [(s); (s)] for 0 < s < t. Its support is given by " inf 0<s<t (s); sup 0<s<t (s) # : (1.10) In particular, the support is always an interval. The support is unbounded if or are unbounded near 0. J. Approx. Theory 99 (1999), 167197. 5 Remark 1.6 Theorem 1.4 has an obvious extension to polynomials that are orthogonal with respect to a discrete measure supp...
Advanced Determinant Calculus
, 1999
"... The purpose of this article is threefold. First, it provides the reader with a few useful and efficient tools which should enable her/him to evaluate nontrivial determinants for the case such a determinant should appear in her/his research. Second, it lists a number of such determinants that have ..."
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Cited by 37 (0 self)
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The purpose of this article is threefold. First, it provides the reader with a few useful and efficient tools which should enable her/him to evaluate nontrivial determinants for the case such a determinant should appear in her/his research. Second, it lists a number of such determinants that have been already evaluated, together with explanations which tell in which contexts they have appeared. Third, it points out references where further such determinant evaluations can be found.
New infinite families of exact sums of squares formulas, Jacobi elliptic functions, and Ramanujan’s tau function
, 1996
"... Dedicated to the memory of GianCarlo Rota who encouraged me to write this paper in the present style Abstract. In this paper we derive many infinite families of explicit exact formulas involving either squares or triangular numbers, two of which generalize Jacobi’s 4 and 8 squares identities to 4n ..."
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Cited by 35 (1 self)
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Dedicated to the memory of GianCarlo Rota who encouraged me to write this paper in the present style Abstract. In this paper we derive many infinite families of explicit exact formulas involving either squares or triangular numbers, two of which generalize Jacobi’s 4 and 8 squares identities to 4n 2 or 4n(n + 1) squares, respectively, without using cusp forms. In fact, we similarly generalize to infinite families all of Jacobi’s explicitly stated degree 2, 4, 6, 8 Lambert series expansions of classical theta functions. In addition, we extend Jacobi’s special analysis of 2 squares, 2 triangles, 6 squares, 6 triangles to 12 squares, 12 triangles, 20 squares, 20 triangles, respectively. Our 24 squares identity leads to a different formula for Ramanujan’s tau function τ(n), when n is odd. These results, depending on new expansions for powers of various products of classical theta functions, arise in the setting of Jacobi elliptic functions, associated continued fractions, regular Cfractions, Hankel or Turánian determinants, Fourier series, Lambert series, inclusion/exclusion, Laplace expansion formula for determinants, and Schur functions. The Schur function form of these infinite families of identities are analogous to the ηfunction identities of Macdonald. Moreover, the powers 4n(n + 1), 2n 2 + n, 2n 2 − n that appear in Macdonald’s work also arise at appropriate places in our analysis. A special case of our general methods yields a proof of the two Kac–Wakimoto conjectured identities involving representing